Document Open Access Logo

The Trisection Genus of Standard Simply Connected PL 4-Manifolds

Authors Jonathan Spreer, Stephan Tillmann

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2018.71.pdf
  • Filesize: 1.2 MB
  • 13 pages

Document Identifiers

Author Details

Jonathan Spreer
Stephan Tillmann

Cite AsGet BibTex

Jonathan Spreer and Stephan Tillmann. The Trisection Genus of Standard Simply Connected PL 4-Manifolds. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 71:1-71:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.71

Abstract

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. In this note we show that the K3 surface has trisection genus 22. This implies that the trisection genus of all standard simply connected PL 4-manifolds is known. We show that the trisection genus of each of these manifolds is realised by a trisection that is supported by a singular triangulation. Moreover, we explicitly give the building blocks to construct these triangulations.
Keywords
  • combinatorial topology
  • triangulated manifolds
  • simply connected 4-manifolds
  • K3 surface
  • trisections of 4-manifolds

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Biplab Basak and Jonathan Spreer. Simple crystallizations of 4-manifolds. Adv. in Geom., 16(1):111-130, 2016. arXiv:1407.0752 [math.GT]. URL: http://dx.doi.org/10.1515/advgeom-2015-0043.
  2. M. Bell, J. Hass, J. Hyam Rubinstein, and S. Tillmann. Computing trisections of 4-manifolds. ArXiv e-prints, nov 2017. URL: http://arxiv.org/abs/1711.02763.
  3. Bruno Benedetti and Frank H. Lutz. Random discrete Morse theory and a new library of triangulations. Exp. Math., 23(1):66-94, 2014. Google Scholar
  4. Benjamin A. Burton and Ryan Budney. A census of small triangulated 4-manifolds. in preparation, ≥2017. Google Scholar
  5. Benjamin A. Burton, Ryan Budney, William Pettersson, et al. Regina: Software for low-dimensional topology. http://regina-normal.github.io/, 1999-2017. Google Scholar
  6. Stewart S. Cairns. Introduction of a Riemannian geometry on a triangulable 4-manifold. Ann. of Math. (2), 45:218-219, 1944. URL: http://dx.doi.org/10.2307/1969264.
  7. Stewart S. Cairns. The manifold smoothing problem. Bull. Amer. Math. Soc., 67:237-238, 1961. URL: http://dx.doi.org/10.1090/S0002-9904-1961-10588-0.
  8. S. K. Donaldson. An application of gauge theory to four-dimensional topology. J. Differential Geom., 18(2):279-315, 1983. URL: http://projecteuclid.org/getRecord?id=euclid.jdg/1214437665.
  9. P. Feller, M. Klug, T. Schirmer, and D. Zemke. Calculating the homology and intersection form of a 4-manifold from a trisection diagram. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1711.04762.
  10. M. Ferri, C. Gagliardi, and L. Grasselli. A graph-theoretical representation of PL-manifolds - a survey on crystallizations. Aequationes Math., 31(2-3):121-141, 1986. URL: http://dx.doi.org/10.1007/BF02188181.
  11. M. Freedman. The topology of four-dimensional manifolds. J. Differential Geom., 17:357-453, 1982. Google Scholar
  12. M. Furuta. Monopole equation and the 11/8-conjecture. Math. Res. Lett., 8(3):279-291, 2001. URL: http://dx.doi.org/10.4310/MRL.2001.v8.n3.a5.
  13. David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol., 20(6):3097-3132, 2016. URL: http://dx.doi.org/10.2140/gt.2016.20.3097.
  14. David T. Gay. Trisections of Lefschetz pencils. Algebr. Geom. Topol., 16(6):3523-3531, 2016. URL: http://dx.doi.org/10.2140/agt.2016.16.3523.
  15. A. Hatcher. Algebraic Topology. Cambridge University Press, 2002. URL: http://books.google.de/books?id=BjKs86kosqgC.
  16. Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11):4983-4997, 2016. URL: http://dx.doi.org/10.1090/proc/13105.
  17. Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11):4983-4997, 2016. URL: http://dx.doi.org/10.1090/proc/13105.
  18. Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in S⁴. Trans. Amer. Math. Soc., 369(10):7343-7386, 2017. URL: http://dx.doi.org/10.1090/tran/6934.
  19. Jeffrey Meier and Alexander Zupan. Genus-two trisections are standard. Geom. Topol., 21(3):1583-1630, 2017. URL: http://dx.doi.org/10.2140/gt.2017.21.1583.
  20. John Milnor and Dale Husemoller. Symmetric bilinear forms, volume 73 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York, 1973. Google Scholar
  21. Mario Pezzana. Sulla struttura topologica delle varietà compatte. Ati Sem. Mat. Fis. Univ. Modena, 23(1):269-277 (1975), 1974. Google Scholar
  22. V. A. Rohlin. New results in the theory of four-dimensional manifolds. Doklady Akad. Nauk SSSR (N.S.), 84:221-224, 1952. Google Scholar
  23. J. Hyam Rubinstein and S. Tillmann. Multisections of piecewise linear manifolds. ArXiv e-prints, 2016. URL: http://arxiv.org/abs/1602.03279.
  24. Nikolai Saveliev. Lectures on the topology of 3-manifolds. De Gruyter Textbook. Walter de Gruyter &Co., Berlin, revised edition, 2012. An introduction to the Casson invariant. Google Scholar
  25. Martin Tancer. Recognition of collapsible complexes is np-complete. Discrete & Computational Geometry, 55(1):21-38, Jan 2016. URL: http://dx.doi.org/10.1007/s00454-015-9747-1.
  26. J. H. C. Whitehead. On C¹-complexes. Ann. of Math. (2), 41:809-824, 1940. URL: http://dx.doi.org/10.2307/1968861.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact